Green Operators in the Edge Calculus

Parameter-dependent Edge Operators

We study parameter-dependent operators on a manifold with edge and construct new classes of elliptic elements in the corner calculus on an infinite cone with a singular base

A Generalized Nonlocal Calculus with Application to the Peridynamics Model for Solid Mechanics

A nonlocal vector calculus was introduced in [2] that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A generalization is developed that provides a more general setting for the nonlocal vector calculus that is independent of particular nonlocal models. It is shown that general nonlocal calculus operators are integral operators with specific integral kernels. General nonlocal calculus properties are developed, including nonlocal integration by parts formula and Green's identities. The nonlocal vector calculus introduced in [2] is…

Nonsmooth Pseudodifferential Boundary Value Problems on Manifolds

We study pseudodifferential boundary value problems in the context of the Boutet de Monvel calculus or Green operators, with nonsmooth coefficients on smooth compact manifolds with boundary. In order to have a definition that is independent of the choice of (smooth) coordinates, we prove that nonsmooth Green operators are invariant under smooth coordinate transformations.

截断分数阶梯度的非局部格林定理与亥姆霍兹分解

Nonlocal Green theorems and Helmholtz decompositions for truncated fractional gradients

本文进一步发展具有有限相互作用支撑域的奇异分数阶型算子相关的非局部微积分理论，推导向量恒等式、分部积分型定理与Helmholtz-Hodge分解，为该框架的后续理论研究与数值实现奠定基础。

迹的理论与散度定理

A theory of traces and the divergence theorem

本文提出一种将迹视为函数空间上线性连续泛函的一般理论，发展出无需法向场和边界迹函数的积分演算，从而在任意Borel集上导出Gauss-Green公式，并应用于证明边值问题弱解的存在性。

Edge-degenerate families of $Ψ$Do's on an infinite cylinder

We establish a parameter-dependent pseudo-differential calculus on an infinite cylinder, regarded as a manifold with conical exits to infinity. The parameters are involved in edge-degenerate form, and we formulate the operators in terms of operator-valued amplitude functions.

Pseudo-differential operators, heat calculus and index theory of groupoids satisfying the Lauter-Nistor condition

In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the calculus of uniformly supported pseudo-differential operators that is analogous to the calculus with bounds defined on manifolds with boundary. We derive a Fredholmness criterion for operators on the Bruhat sphere, and prove that their parametrices up to compact operators lie inside…

Boundary singularity of Poisson and harmonic Bergman kernels

We give a complete description of the boundary behaviour of the Poisson kernel and the harmonic Bergman kernel of a bounded domain with smooth boundary, which in some sense is an analogue of the similar description for the usual Bergman kernel on a strictly pseudoconvex domain due to Fefferman. Our main tool is the Boutet de Monvel calculus of pseudodifferential boundary operators, and in fact we describe the boundary singularity of a general potential, trace or singular Green operator from that calculus.

Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

Let M denote the space of probability measures on R^D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced by Ambrosio, Gigli and Savare'. In this paper we develop a calculus for the corresponding class of differential forms on M. In particular we prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For D=2d we then define a symplectic distribution on M in terms of this calculus, thus obtaining a rigorous framework for the notion…

多面体与多面锥上时间可测抛物型算子的格林函数估计

Green's function estimates for time measurable parabolic operators on polyhedrons and polyhedral cones

本文在三维多面体与多面锥区域上建立抛物型算子的格林函数估计，引入依赖顶点、边和边界距离的混合权重，并由区域几何明确给出权重参数的允许范围。

Spectral asymptotics for nonsmooth singular Green operators

Singular Green operators G appear typically as boundary correction terms in resolvents for elliptic boundary value problems on a domain Ω\subset R^n, and more generally they appear in the calculus of pseudodifferential boundary problems. In particular, the boundary term in a Krein resolvent formula is a singular Green operator. It is well-known in smooth cases that when G is of negative order -t on a bounded domain, its eigenvalues or s-numbers have the behavior (*) s_j(G) \sim c j^{-t/(n-1)} for j\to \infty, governed by the boundary dimension n-1. In some nonsmooth cases, upper estimates (**)…

带全局投影条件的参数化边值问题的微积分

Calculus for parametric boundary problems with global projection conditions

本文构造了一种处理带边光滑流形上参数依赖算子的伪微分微积分，包含全局投影边界条件下的预解式，并通过三个主符号的可逆性刻画参数椭圆性。

有限元外微积分在无坐标单纯复形上的Rust实现

Rust Implementation of Finite Element Exterior Calculus on Coordinate-Free Simplicial Complexes

本论文开发了一个基于有限元外微积分的Rust库，用于在任意维无坐标单纯复形上求解Hodge-Laplace问题，采用Regge度量与一阶Whitney基函数，并通过收敛性研究部分验证。

具有复杂权重的网络：格林函数和幂级数

Networks with complex weights: Green function and power series

本文研究了具有复数权重的有限和无限网络中的格林函数及其相关核，通过与可逆马尔可夫链的比较，展示了复数权重下的瞬态和复发性概念的推广。